Spectral asymptotics and Lamé spectrum for coupled particles in periodic potentials

TL;DR

The paper analyzes the stability of synchronous motion in a pair of elastically coupled particles in a $2\pi$-periodic potential $V$. It reveals that the linearized Floquet spectrum around the synchronous solution exhibits an asymptotic periodicity in $\ln E$ as the energy $E\to 0$, and derives a normal-form reduction giving $\mathrm{tr}\,F_E = a\cos\left(\frac{\omega}{\lambda}\ln E - \varphi\right) + o(1)$ with $\lambda=\sqrt{-V''(\pi)}$ and $\omega^2=2\kappa+\lambda^2$; topological arguments show infinitely many stability-instability crossings accumulating at $E=0$. For large energies, the synchronous motion is linearly stable for any $C^2$ periodic potential, with a short-time expansion of the Floquet map ruling out real eigenvalues. In the special sinusoidal case $V(x)=\kappa\cos x$, the linearization reduces to Lamé's equation with $n=1$, leading to a collapsed resonance structure where only one instability interval remains open, explicitly between $E_2$ and $E_1$ with $E_1=4\kappa$ and $E_2=4\kappa-2$ (for suitable $\kappa$). The results connect a basic coupled-particle model to finite-gap (Lamé) theory, illustrating how classic elliptic-function spectral problems arise in the simplest dynamical setting and clarifying the stability landscape across energy scales.

Abstract

We make two observations on the motion of coupled particles in a periodic potential. Coupled pendula, or the space-discretized sine-Gordon equation is an example of this problem. Linearized spectrum of the synchronous motion turns out to have a hidden asymptotic periodicity in its dependence on the energy; this is the gist of the first observation. Our second observation is the discovery of a special property of the purely sinusoidal potentials: the linearization around the synchronous solution is equivalent to the classical Lamè equation. As a consequence, {\it all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials}. This provides a new example where Lamé's finite zone potential arises in the simplest possible setting.

Figures (5)

• Figure 1: Elastically coupled particles in a periodic potential $V$. The case $V= \cos$ corresponds to torsionally coupled pendula.
• Figure 2: Periodic potential $V$.
• Figure 3: For $V(x)= \cos x$ all but one resonance intervals collapse; for $V(x) = \cos ^3x$ the intervals open up.
• Figure 4: Proof of (\ref{['eq:p']}).
• Figure 5: The path of the Floquet matrix in $SL(2, {\mathbb R} )$ for a generic potential as a function of the eigenvalue parameter $\lambda$ (left) and for the Lamé potential as the function of the energy $E$ (right). All the $E$--intervals except for $[E_2, E_1]$ are collapsed.

Theorems & Definitions (16)

• Theorem 1
• proof
• Theorem 2
• Lemma 1
• Lemma 2: "exponential death"
• Lemma 3
• proof
• proof : Proof of Lemma \ref{['T']}
• proof : Proof of Lemma \ref{['asyconst']}
• proof : Proof of Lemma \ref{['proxy']}